Where Rough Sets and Fuzzy Sets Meet

• Autorzy: Polkowski L. , Semeniuk-Polkowska M.

Identyfikatory

DOI

10.3233/FI-2015-1294

• Języki publikacji: EN

Abstrakty

EN

The title of this note announces an attempt at pointing to particular points where rough set and fuzzy set approaches to decision making are close to each other, the closeness meaning that formal approaches to behaviour of the two at those points can be given in an analogous form. This by no means implies that the two can be unified as theories dealing with uncertainty. As the notion of truth for rough set decision rules is well established, we propose a notion of truth for fuzzy decision rules and we seek an analogy between the two. In order to introduce an analogous form of graded notion of truth for decision rules in both theories, we introduce a new context in which to set this notion. This context is based on our earlier results concerning rough mereological granular logics and their relevance for rough decision rules. To make our exposition satisfactorily complete, we recall our approach to granularity based on rough mereology. This note is partitioned into two parts. In Prologue, four sections present basic ideas on vagueness and ambiguity, rough sets, fuzzy sets and mereology along with rough mereology. Here also notions of truth for rough and fuzzy decision rules are presented. In Episode, main protagonists enter an analysis aimed at pointing to further close analogies between them, notably concerning notions of partial truth and dependency among attributes. To this end in four sections on similarity, granulation of knowledge, granular logics and dependencies,we give basic information on similarity, granulation and dependency and we point to analogies between the two theories with respect to those notions. We sum up the content of the note in Conclusions.

Słowa kluczowe

EN

fuzzy sets; rough sets; decision theory; protagonists; decision making

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2015
• Tom: Vol. 142, nr 1-4
• Strony: 269--284
• Opis fizyczny: Bibliogr. 44 poz.

Twórcy

autor

Polkowski L.

• Polish–Japanese Institute of Information Technology Koszykowa 86, 02008 Warsaw, Poland
• Department of Mathematics and Computer Science University of Warmia and Mazury Słoneczna str. 54, 10560 Olsztyn, Poland

autor

Semeniuk-Polkowska M.

• Chair of Formal Linguistics, Warsaw University Browarna 8/12, 00650 Warsaw,Poland

Bibliografia

• [1] Van Benthem, J.: A Manual of Intensional Logic. CSLI Stanford University, Stanford, CA (1988).
• [2] Black, M.: Vagueness. An exercise in logical analysis. Philosophy of Science 4(4), pp 427–455 (1937).
• [3] Cornelis, C., Jensen, R., Hurtado Martin, G., ´Sle¸zak, D.: Attribute selection with fuzzy decision rules. Information Sciences 180(2), pp 209–224 (2010).
• [4] Fleck, L.: Some specific features of the medical way of thinking (in Polish). Archiwum Historii i Filozofii Medycyny oraz Historii Nauk Przyrodniczych 6, pp. 55–64 (1927); also: Cohen,R.S., Schnelle, Th. (eds.). Cognition and Fact. Materials on Ludwik Fleck. Reidel, Dordrecht, pp 445-457 (1986).
• [5] Frege, G.: Grundgesetze der Arithmetik. Verlag Hermann Pohle, Jena (1903).
• [6] Hájek, P.: Metamathematics of Fuzzy Logic. Kluwer, Dordrecht (2001).
• [7] Hájek, P.: On vagueness, truth values and fuzzy logic.Studia Logica 91(3), pp 367–382 (2009).
• [8] Kuratowski, C., Mostowski, A.: Set Theory.Polish Scientific Publishers and North Holland.Warszawa, Amsterdam (1967).
• [9] Leśniewski, S.: On the foundations of set theory. Topoi 2, pp 7–52 (1982)(a translation of the original: O Podstawach Teoryi Zbiorow (in Polish), Moscow, 1916).
• [10] Łukasiewicz, J.: Die Logischen Grundlagen derWahrscheinlichtkeitsrechnung. Cracow (1913).
• [11] Marczewski, E.: Independence d’ensembles et prolongement de mesures. Colloq. Math. 1, pp 122–132 (1948).
• [12] Menger, K.: Statistical metrics. Proc. Natl. Ac. Sci. 28 (1942).
• [13] Menger, K.: Ensembles flous et fonctions aleatoires. C.R. Paris 232, pp 2001–2003 (1951).
• [14] Montague, R.: Thomason, R. (ed.). Formal Philosophy, Yale University Press, New Haven (1974).
• [15] Nieminen, J.: Rough tolerance equality and tolerance black boxes. Fundamenta Informaticae 11, pp 289–296 (1988).
• [16] Novotny M., Pawlak Z.: Partial dependency of attributes. Bull. Pol. Ac.: Math. 36, pp 453–458 (1988).
• [17] Pal, S. K., Mitra, P.: Case generation using rough sets with fuzzy representation. IEEE Transactions on Knowledge and Data Engineering 16(3), pp 293–300 (2004).
• [18] Pal, S. K., Skowron, A.: Rough–Fuzzy Hybridization: A New Trend in Decision Making. Springer–Verlag New York, Secausus NJ (1999).
• [19] Pawlak, Z: Rough sets. Intern. J. Comp. Inform. Sci. 11, pp 341–366 (1982).
• [20] Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer, Dordrecht (1991).
• [21] Pawlak, Z.: Vagueness and uncertainty. A rough set perspective. ICS Res. report 19/94. Warsaw University of Technology. Institute of Computer Science, Warsaw (1994).
• [22] Pedrycz,W., Gomide, F.: An Introduction to Fuzzy Sets. MIT Press, Cambridge MA (1998).
• [23] Polkowski, L. : Mereology and uncertainty. Logic and Logical Philosophy, to appear.
• [24] Polkowski, L.: Mereology in engineering and computer science. In: Calosi,C., Graziani, P. (eds.).Mereology and the Sciences. Parts and Wholes in the Contemporary Scientific Context. Synthese Library vol. 371, Springer Verlag, Berlin, pp 217–292 (2014).
• [25] Polkowski, L.: Approximate reasoning by Parts. An Introduction to Rough Mereology. Springer Verlag, Berlin, 2011.
• [26] Polkowski, L.: Rough mereology in analysis of vagueness. In: Proceedings RSKT 2008, Lecture Notes in Artificial Intelligence 5009, pp. 197–205 (2008).
• [27] Polkowski, L.: A note on 3–valued rough logic accepting decision rules. Fundamenta Informaticae 61, pp. 37–45 (2004).
• [28] Polkowski, L.: Formal granular calculi based on rough inclusions (a feature talk). In: Proceedings IEEE GrC 2005, pp 57–62, IEEE Press, Piscataway, NJ (2005).
• [29] Polkowski, L.: A unified approach to granulation of knowledge and granular computing based on roughmereology: a survey. In: Pedrycz, W., Skowron, A., Kreinovich, V., (eds.). Handbook of Granular Computing, John Wiley and Sons Ltd., Chichester UK, pp. 375–400 (2008).
• [30] Polkowski, L.: Granulation of Knowledge: Similarity Based Approach in Information and Decision Systems. In: Meyers, R.A. (ed.). Springer Encyclopedia of Complexity and System Sciences, article 00708 (2009).
• [31] Polkowski, L., Artiemjew, P.: Granular Computing in Decision Approximation. An Application of Rough Mereology, Springer Verlag, ISRL vol. 77, Berlin, release as of Feb. 14, 2015.
• [32] Polkowski L., Semeniuk–Polkowska M.: On rough set logics based on similarity relations. Fundamenta Informaticae 64, pp 379–390 (2005).
• [33] Polkowski, L., Skowron, A.: Rough mereology. In: Proceedings ISMIS’94. Lecture Notes in Artificial Intelligence 869, pp 85–94 (1994).
• [34] Polkowski, L., Skowron, A., Zytkow, J.: Tolerance based rough sets. In: Lin, T.Y.,Wildberger,M.: Soft Computing: Rough sets, fuzzy logic, neural networks, uncertainty management, knowledge discovery. Simulation Councils Inc., San Diego, pp 55–58 (1994).
• [35] Ruspini E. H.: On the semantics of fuzzy logic. Int. J. Approx. Reasoning 5, pp 45 – 88 (1991).
• [36] Sennet, A.: Ambiguity.In: Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/ambiguity/
• [37] Shreider Yu.: Equality, Resemblance, Order. Mir Publishers, Moscow (1960).
• [38] Skowron, A., Stepaniuk, J.: Tolerance approximation spaces. Fundamenta Informaticae 27, pp 245–253 (1996).
• [39] Sorensen, R.: Vagueness. In: Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/vagueness/
• [40] Varzi, A. C.: Mereology. In: Stanford Encyclopedia of Philosophy. Available: http://plato.stanford.edu/entries/mereology/.
• [41] Zadeh, L.A.: Fuzzy sets. Information and Control 8, pp 338 – 353 (1965).
• [42] Zadeh, L.A.: Similarity relations and fuzzy orderings. Information Sciences. An International Journal 3(2), pp 177–200 (1971).
• [43] Zadeh, L. A.: Fuzzy sets and information granularity. In: Gupta, M., Ragade, R., Yager, R.R., (eds.). Advances in Fuzzy Set Theory and Applications, North–Holland, Amsterdam , pp 3–18 (1979).
• [44] Ziarko,W.: Variable precision rough set model. J. Computer and System Sciences 46(1), pp 39–59 (1993